Records of arrays are used to measure wavefields and extract information about the medium through which the wavefield propagates. In seismology, array analyses have been applied in the estimation of seismic scatter distribution, measurements of surface wave dispersion, measurements of seismic wavefields associated with volcanic activity, estimation of earthquake rupture propagation, and other analyses.
Techniques of array analysis generally may be divided into two categories, depending on whether they operate in a frequency domain or in a time domain. In particular, frequency domain approaches have the capability of resolving multiple signals arriving simultaneously at an array due to multipathing or use of multiple sources.
One of the simplest frequency-wavenumber methods is the beamforming or conventional method. The estimated power spectrum can be shown to be a convolution of the array response with the true power spectrum. Here the array response is determined completely by the number of sensors and spatial distribution. Because the array response has a broad beam width for the incident signal""s wavenumber and side lobes, the power spectrum estimated by the conventional method is blurred by the array response.
The minimum variance distortionless method has been shown to produce significantly higher resolutions for a single identified wave than the resolution produced by the conventional method. However, the resolving power of the minimum variance distortionless method is similar to that of the conventional method when multiple waves are present. For the resolution of multiple signals, a signal subspace approach was proposed, referred to as multiple signal characterization (MUSIC) (see, e.g., Johnson and Dudgeon Prentice-Hall Signal Processing Series, ISBN 0-13-048513-6). Key to the performance of MUSIC is the division of information in the spatial cross-covariance matrix of the Fourier coefficients into two vector subspacesxe2x80x94a signal subspace and a noise subspace. The MUSIC approach allows for high-resolution estimates of the power spectrum since it does not convolve the true spectrum with the array response.
Methods to generate high-resolution frequency-wavenumber spectra and higher dimension extension frequency-wavenumber-wavenumber spectra are needed in the art of seismic analysis. The methods of the present invention satisfy this need.
In seismic data analysis, using a small number of sensors to generate high-resolution frequency-wavenumber (fk) spectra and higher dimension extension frequency-wavenumber-wavenumber (fkk) spectra is highly desirable and has wide-ranging applications. Such applications include local velocity measurements and the detection of dispersive waves. When relatively few samples are available, parametric methods provide a superior wavenumber resolution compared to a standard fk-transform. Parametric methods such as the multiple signal classification (MUSIC) method can be applied to wavenumber-wavenumber or wavenumber estimation at each frequency in order to generate the algorithms fkk-MUSIC and fk-MUSIC and many other parametric algorithms for use in seismic applications. In addition, parametric methods may be extended to fkkk or sparse/irregular arrays. The techniques of the present invention prove to be useful tools whenever estimating local velocities involves data from a limited number of receivers.
Thus, the present invention provides a method of seismic analysis comprising a temporal Fourier transform and a spatial estimation based on a parametric algorithm. In one embodiment of the present invention, the parametric algorithm is MUSIC. In one aspect of this embodiment of the present invention, the parametric algorithm is applied to data from equidistant spatial sampling. In another aspect of this embodiment of the present invention, the parametric algorithm is applied to data from sparse spatial sampling. In this embodiment of the invention, the data is fed into the parametric algorithm as an MNxc3x97L matrix, wherein MN is the number of sensors in a two dimensional array and L is the number of samples per sensor. Also in this embodiment of the present invention, the Fourier transform is implemented on each of the traces, and the parameters passed to the parametric algorithm are selected from the group of order of Fourier transform, sampling frequency, distance between receivers in an x direction, distance between receivers in a y direction, number_of_sources, or f_span.
Alternative embodiments of the present invention apply other parametric methods including, but not limited to, the Minimum Variance algorithm, Eigenvector algorithm, Maximum Likelihood algorithm, Pisarenco algorithm, the ARMA (Autoregressive Moving Average Algorithm), AR or MA algorithms and the Maximum Entropy algorithm.